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Differentially Private E-Values

Daniel Csillag, Diego Mesquita

TL;DR

This work addresses the tension between privacy and validity in e-value-based inference by introducing a general framework that privatizes e-values with biased multiplicative noise: $E^{\mathrm{DP}}(D) = E(D) \cdot e^{-{\xi}}$. By carefully designing the noise $\xi$ (e.g., biased Gaussian or biased Laplace) to satisfy a moment-generating function constraint, the authors achieve $(\alpha,\epsilon)$-Rényi DP while preserving the fundamental e-value validity and post-hoc properties. They provide an exact characterization of the power loss through $\mathbb{E}[\frac{1}{n}\log E^{\mathrm{DP}}(D)] = \mathbb{E}[\frac{1}{n}\log E(D)] - \mathbb{E}[\xi]/n$, which vanishes as $n$ grows, ensuring asymptotic parity with non-private e-values. The framework maintains compositionality and supports DP versions of confidence intervals and e-conformal predictions, with experiments in private healthcare, online risk monitoring, and phishing-detection demonstrating practical impact. Overall, this work enables private, anytime-valid, e-value-based inference suitable for sensitive data domains.

Abstract

E-values have gained prominence as flexible tools for statistical inference and risk control, enabling anytime- and post-hoc-valid procedures under minimal assumptions. However, many real-world applications fundamentally rely on sensitive data, which can be leaked through e-values. To ensure their safe release, we propose a general framework to transform non-private e-values into differentially private ones. Towards this end, we develop a novel biased multiplicative noise mechanism that ensures our e-values remain statistically valid. We show that our differentially private e-values attain strong statistical power, and are asymptotically as powerful as their non-private counterparts. Experiments across online risk monitoring, private healthcare, and conformal e-prediction demonstrate our approach's effectiveness and illustrate its broad applicability.

Differentially Private E-Values

TL;DR

This work addresses the tension between privacy and validity in e-value-based inference by introducing a general framework that privatizes e-values with biased multiplicative noise: . By carefully designing the noise (e.g., biased Gaussian or biased Laplace) to satisfy a moment-generating function constraint, the authors achieve -Rényi DP while preserving the fundamental e-value validity and post-hoc properties. They provide an exact characterization of the power loss through , which vanishes as grows, ensuring asymptotic parity with non-private e-values. The framework maintains compositionality and supports DP versions of confidence intervals and e-conformal predictions, with experiments in private healthcare, online risk monitoring, and phishing-detection demonstrating practical impact. Overall, this work enables private, anytime-valid, e-value-based inference suitable for sensitive data domains.

Abstract

E-values have gained prominence as flexible tools for statistical inference and risk control, enabling anytime- and post-hoc-valid procedures under minimal assumptions. However, many real-world applications fundamentally rely on sensitive data, which can be leaked through e-values. To ensure their safe release, we propose a general framework to transform non-private e-values into differentially private ones. Towards this end, we develop a novel biased multiplicative noise mechanism that ensures our e-values remain statistically valid. We show that our differentially private e-values attain strong statistical power, and are asymptotically as powerful as their non-private counterparts. Experiments across online risk monitoring, private healthcare, and conformal e-prediction demonstrate our approach's effectiveness and illustrate its broad applicability.
Paper Structure (23 sections, 33 theorems, 69 equations, 3 figures, 4 tables)

This paper contains 23 sections, 33 theorems, 69 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Background
  4. Differential privacy
  5. E-values
  6. Differentially Private E-Values
  7. Hypothesis testing
  8. Algorithms atop e-values and confidence intervals
  9. Experiments and Applications
  10. Private e-confidence intervals
  11. Anytime-valid hypothesis testing for online risk monitoring
  12. Private e-conformal prediction
  13. Conclusion
  14. Proofs from the main text
  15. Additional theoretical results
  16. ...and 8 more sections

Key Result

Proposition 2.3

For any $\alpha > 1$, let algorithms $\mathcal{A}_1(\cdot)$ and $\mathcal{A}_2(\cdot)$ be $(\alpha, \epsilon_1)$- and $(\alpha, \epsilon_2)$-Rényi differentially private. Then the algorithm $\mathcal{A}(\cdot) = (\mathcal{A}_1(\cdot), \mathcal{A}_2(\cdot))$ is $(\alpha, \epsilon_1 + \epsilon_2)$-Rén

Figures (3)

  • Figure 1: Private e-confidence intervals. We use our differentially private e-values to produce private confidence intervals for the prevalence of diabetes over a population. We illustrate the confidence intervals varying the number of samples $n$ (x axis) and Rényi privacy budgets $(\alpha, \epsilon)$. We display the non-private confidence intervals (blue) along with those obtained through our biased Gaussian mechanism (orange), and when applicable also the as the ones obtained via our biased Laplace mechanism (green). Note that our biased Gaussian mechanism always converges to the non-private confidence intervals, as does the biased Laplace mechanism when available.
  • Figure 2: Anytime-valid private hypothesis testing. We apply our differentially private e-values for private continuous risk monitoring by the means of anytime-valid hypothesis testing. Our goal is to detect significant increases in the risk of a deployed predictive model, whose moving average can be seen in the background. The curves indicate the log of the e-values for the null hypothesis that the risk is under a certain safety threshold. Our private e-values closely match the non-private ones, quickly rejecting the null hypothesis after the change-point even for very small values of $\epsilon$.
  • Figure 3: Private e-conformal prediction. We also apply our e-values to produce private predictive sets with post-hoc guarantees through e-conformal prediction. In this setting the biased Laplace mechanism is generally applicable, and consistently outperforms the biased Gaussian mechanism, closely matching the average size of non-private prediction sets.

Theorems & Definitions (52)

  • Definition 2.1: $(\epsilon, \delta)$-differential privacy
  • Definition 2.2: $(\alpha, \epsilon)$-Rényi differential privacy
  • Proposition 2.3: Composition of $(\alpha, \epsilon)$-Rényi differential privacy
  • Proposition 2.4: Post-processing of $(\alpha, \epsilon)$-Rényi differential privacy
  • Definition 2.5: E-value
  • Proposition 2.6: E-to-p conversion; post-hoc validity
  • Proposition 2.7: Optional continuation
  • Proposition 2.8: Averaging
  • Theorem 3.1: Biased Gaussian mechanism
  • Theorem 3.2: Biased Laplace mechanism
  • ...and 42 more